LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

2.4 Linear Functions Graphing Linear Functions Standard Form Ax + By = C Slope Average Rate of Change Linear Models

Linear Function A function  is a linear function if, for real numbers a and b,

Graph (x) = – 2x + 6. Give the domain and range. GRAPHING A LINEAR FUNCTION USING INTERCEPTS Example 1 Graph (x) = – 2x + 6. Give the domain and range. Solution The x-intercept is found by letting (x) = 0 and solving for x. Add 2x; divide by 2.

Graph (x) = – 2x + 6. Give the domain and range. GRAPHING A LINEAR FUNCTION USING INTERCEPTS Example 1 Graph (x) = – 2x + 6. Give the domain and range. Solution The x-intercept is 3, so we plot (3, 0). The y-intercept is Plot this point and connect the two points with a straight line. Find a check point.

Graph (x) = – 2x + 6. Give the domain and range. GRAPHING A LINEAR FUNCTION USING INTERCEPTS Example 1 Graph (x) = – 2x + 6. Give the domain and range. y Solution (0, 6) check point y-intercept (2, 2) x (3, 0) The domain and range are both(– , ). x-intercept

Graph (x) = – 3. Give the domain and range. GRAPHING A HORIZONTAL LINE Example 2 Graph (x) = – 3. Give the domain and range. y Solution Since (x), or y, always equals – 3, the value of y can never be 0. A line with no x-intercept is parallel to the x-axis. The domain is (– , ). The range is {– 3}. x Horizontal line (0, – 3)

Graph x = – 3. Give the domain and range. GRAPHING A VERTICAL LINE Example 3 Graph x = – 3. Give the domain and range. y Solution Since x always equals – 3, the value of x can never be 0, and the graph has no y-intercept and is parallel to the y-axis Vertical line ( – 3, 0) x

Graph x = – 3. Give the domain and range. GRAPHING A VERTICAL LINE Example 3 Graph x = – 3. Give the domain and range. y The domain of this relation, which is not a function, is {– 3}. The range is (– , ). Vertical line ( – 3, 0) x

Slope An important characteristic of a straight line is its slope, a numerical measure of the steepness of a line. Geometrically it may be interpreted as the ratio of rise to run. Use two distinct points. The change in the horizontal distance, x2 – x1, is denoted as ∆x (delta x) and the change in the vertical distance, y2 – y1, is denoted as ∆y.

Slope The slope m of a line through points (x1, y1) and (x2, y2) is where ∆x ≠ 0.

Caution When using the slope formula, it makes no difference which point is used (x1, y1) or (x2, y2); however, be consistent . Start with the x- and y-values of one point (either one) and subtract the corresponding values of the other point. Be sure to write the difference of the y-values in the numerator and the difference of the x-values in the denominator.

Undefined Slope The slope of a vertical line is undefined.

Find the slope of the line through the given points. FINDING SLOPES WITH THE SLOPE FORMULA Example 5 Find the slope of the line through the given points. b. Solution Undefined The slope of a vertical line is undefined.

Find the slope of the line through the given points. FINDING SLOPES WITH THE SLOPE FORMULA Example 5 Find the slope of the line through the given points. c. Solution Drawing a graph through these two points would produce a horizontal line.

Zero Slope The slope of a horizontal line is 0.

Slopes A line with a positive slope rises from left to right. A line with a negative slope falls from left to right. When the slope is positive, the function is increasing. When the slope is negative, the function is decreasing.

Average Rate of Change We know that the slope of a line is the ratio of the vertical change in y to the horizontal change in x. So, the slope gives the rate of change in y per unit of change in x, where the value of y depends on the value of x. If  is a linear function defined on [a, b], then